Some Properties of the Pivotal Statistic Based on the Asymptotically Distribution-Free Theory in Structural Equation Modeling

For normally distributed data, the asymptotic bias and skewness of the pivotal statistic Studentized by the asymptotically distribution-free standard error are shown to be the same as those given by the normal theory in structural equation modeling. This gives the same asymptotic null distributions of the two pivotal statistics up to the next order beyond the usual normal approximation under normality. With an alternative hypothesis, the asymptotic variances of the two statistics under normality/non normality are also derived. It is, however, shown that the asymptotic variances of the non null distributions of the statistics are generally different even under normality.     

Testing Normality Against the Logistic Distribution Using Saddlepoint Approximation

We consider the problem of testing normality against the logistic distribution, based on a random sample of observations. Since the two families are separate (non nested), the ratio of maximized likelihoods (RML) statistic does not have the usual asymptotic chi-square distribution. We derive the saddlepoint approximation to the distribution of the RML statistic and show that this approximation is more accurate than the normal and Edgeworth approximations, especially for tail probabilities that are the main values of interest in hypothesis testing. It is also shown that this test is almost identical to the most powerful invariant test.     

Improvement of the quality of the chi-square approximation for the ADF test on a covariance matrix with a linear structure

The asymptotically distribution-free (ADF) test statistic was proposed by Browne (1984). It is known that the null distribution of the ADF test statistic is asymptotically distributed according to the chi-square distribution. This asymptotic property is always satisfied, even under nonnormality, although the null distributions of other famous test statistics, e.g., the maximum likelihood test statistic and the generalized least square test statistic, do not converge to the chi-square distribution under nonnormality. However, many authors have reported numerical results which indicate that the quality of the chi-square approximation for the ADF test is very poor, even when the sample size is large and the population distribution is normal. In this paper, we try to improve the quality of the chi-square approximation to the ADF test for a covariance matrix with a linear structure by using the Bartlett correction applicable under the assumption of normality. By conducting numerical studies, we verify that the obtained Bartlett correction can perform well even when the assumption of normality is violated.     

The Moments Do Not Determine a Distribution

Table 1 corrects the critical values for testing normality reported by Lilliefors (1967). The corrected table allows us to derive a simple analytic approximation to the upper tail probabilities of his test statistic for probabilities less than .10. With few exceptions, the approximation is more accurate than Lilliefors's original table.     

On the distribution of a new measure of heaviness of tails

Distributional properties are given for a statistic T*, which has previously been reported to have power properties as a test of normality as attractive as those of the sample kurtosis or perhaps slightly more attractive. Asymptotic results, the mean and variance under normality, the range of variation, and approximation of critical values for testing normality are obtained     

A Simple Derivation of r* for Curved Exponential Families

For curved exponential families we consider modified likelihood ratio statistics of the form r_L=r+ log( u/r)/r , where r is the signed root of the likelihood ratio statistic. We are testing a one-dimensional hypothesis, but in order to specify approximate ancillary statistics we consider the test as one in a series of tests. By requiring asymptotic independence and asymptotic normality of the test statistics in a large deviation region there is a particular choice of the statistic u which suggests itself. The derivation of this result is quite simple, only involving a standard saddlepoint approximation followed by a transformation. We give explicit formulas for the statistic u , and include a discussion of the case where some coordinates of the underlying variable are lattice.     

Multivariate Saddlepoint test for the wrapped normal model

In this article we show the effectiveness and the accuracy of the test statistic based on the expnnent of the saddlepoint approximation for the density of M-estimators, proposed by Robinson, Ronchetti and Young (1999), for testing simultaneous hypotheses on the mean and on the variance of a wrapped normal distribution. We base this test statistic on the trigonometric method of moments estimator proposed by Gatto and Jammalamadaka (l999b), which admits the M-estimator representation necessary for this test. This test statistic has an approximate chi-squared distribution, asympiotically up to the second order, and the high accuracy of this approximation is shown by numerical simulations.     

On comparing several straight lines under heteroscedasticity and robustness with respect to departure from normality

This article considers the problem of testing slopes in k straight lines with'heterogeneous variances. The statistic Fβ is proposed and the null and non-null distributions of Fβ derived under normality assumption. The power function values are then approximated by Laguerre polynomial expansion for normal and non-normal universes. For the example given in Graybill ‘1976, p. 295’, it is shown that the Satterthwaite approximation provides a close approximation to the null and non-null distributions in all the cases; it is also shown that the Fβ test is quite robust with respect to departure from normality in the case of mixtures of two normals.     

Approximation of distributions by using the Anderson Darling statistic

ABSTRACT

In practice, it is often not possible to find an appropriate family of distributions which can be used for fitting the sample distribution with high precision. In these cases, it seems to be opportune to search for the best approximation by a family of distributions instead of an exact fit. In this paper, we consider the Anderson–Darling statistic with plugged-in minimum distance estimator for the parameter vector. We prove asymptotic normality of the Anderson–Darling statistic which is used for a test of goodness of approximation. Moreover, we introduce a measure of discrepancy between the sample distribution and the model class.     

Re-examining two tests for bivariate normality

Many goodness of fit tests for bivariate normality are not rigorous procedures because the distributions of the proposed statistics are unknown or too difficult to manipulate. Two familiar examples are the ring test and the line test. In both tests the statistic utilized generally is approximated by a chi-square distribution rather than compared to its known beta distribution. These two procedures are re-examined and re-evaluated in this paper. It is shown that the chi-square approximation can be too conservative and can lead to unnecessary

rejection of normality.     

Omnibus test of normality using the Q statistic

A new statistical procedure for testing normality is proposed. The Q statistic is derived as the ratio of two linear combinations of the ordered random observations. The coefficients of the linear combinations are utilizing the expected values of the order statistics from the standard normal distribution. This test is omnibus to detect the deviations from normality that result from either skewness or kurtosis. The statistic is independent of the origin and the scale under the null hypothesis of normality, and the null distribution of Q can be very well approximated by the Cornish-Fisher expansion. The powers for various alternative distributions were compared with several other test statistics by simulations.     

Generalized bartlett correction

This paper provides a general method of modifying a statistic of interest in such a way that the distribution of the modified statistic can be approximated by an arbitrary reference distribution to an order of accuracy of O(n ^-1/2) or even O(n ^-1). The reference distribution is usually the asymptotic distribution of the original statistic. We prove that the multiplication of the statistic by a suitable stochastic correction improves the asymptotic approximation to its distribution. This paper extends the results of the closely related paper by Cordeiro and Ferrari (1991) to cope with several other statistical tests. The resulting expression for the adjustment factor requires knowledge of the Edgeworth-type expansion to order O(n^-1) for the distribution of the unmodified statistic. In practice its functional form involves some derivatives of the reference distribution. Certain difference between the cumulants of appropriate order in n of the unmodified statistic and those of its first-order approximation, and the unmodified statistic itself. Some applications are discussed.     

Simulation-assisted saddlepoint approximation

A general saddlepoint/Monte Carlo method to approximate (conditional) multivariate probabilities is presented. This method requires a tractable joint moment generating function (m.g.f.), but does not require a tractable distribution or density. The method is easy to program and has a third-order accuracy with respect to increasing sample size in contrast to standard asymptotic approximations which are typically only accurate to the first order.

The method is most easily described in the context of a continuous regular exponential family. Here, inferences can be formulated as probabilities with respect to the joint density of the sufficient statistics or the conditional density of some sufficient statistics given the others. Analytical expressions for these densities are not generally available, and it is often not possible to simulate exactly from the conditional distributions to obtain a direct Monte Carlo approximation of the required integral. A solution to the first of these problems is to replace the intractable density by a highly accurate saddlepoint approximation. The second problem can be addressed via importance sampling, that is, an indirect Monte Carlo approximation involving simulation from a crude approximation to the true density. Asymptotic normality of the sufficient statistics suggests an obvious candidate for an importance distribution.

The more general problem considers the computation of a joint probability for a subvector of random T, given its complementary subvector, when its distribution is intractable, but its joint m.g.f. is computable. For such settings, the distribution may be tilted, maintaining T as the sufficient statistic. Within this tilted family, the computation of such multivariate probabilities proceeds as described for the exponential family setting.     

A Generalized Distance Multi-Sample Test of Normality With Applications to Process Control

In statistical process control one typically takes periodic small samples. Statistical inferences made from these samples often assume that the samples come from normal distributions with the means and variances possibly changing over time. A multisample test of normality is proposed to test this assumption. The test statistic is the generalized distance between the standardized order statistic vector averaged across the samples and its expected value under normality. The null distribution of the statistic approaches a chi-squared distribution as the number of samples increases. A Monte Carlo study suggests that the test has desirable power properties relative to competing tests.     

Likelihood Based Inference in Non-linear Regression Models Using the p* and R* Approach

We develop second order asymptotic results for likelihood-based inference in Gaussian non-linear regression models. We provide an approximation to the conditional density of the maximum likelihood estimator given an approximate ancillary statistic (the affine ancillary). From this approximation, we derive a statistic to test an hypothesis on one component of the parameter. This test statistic is an adjustment of the signed log-likelihood ratio statistic. The distributional approximations (for the maximum likelihood estimator and for the test statistic) are of second order in large deviation regions.     

An approximation of the distribution function of the LIML identifiability test statistic using the method of moments

This paper contains an application of the asymptotic expansion of a pFp() function to a problem encountered in econometrics. In particular we consider an approximation of the distribution function of the limited information maximum likelihood (LIML) identifiability test statistic using the method of moments. An expression for the Sth order asymptotic approximation of the moments of the LIML identifiability test statistic is derived and tabulated. The exact distribution function of the test statistic is approximated by a member of the class of F (variance ratio) distribution functions having the same first two integer moments. Some tabulations of the approximating distribution function are included.     

Testing for normally in censored regressions

This paper derives a Lagrange Multiplier test for normality in censored regressions. The test is derived against the generalized log-gamma distribution, in which normal is a special case. The resulting test statistic coincides to some extent with previously suggested score and conditional moment tests. Estimation of the variance is performed by using the matrix of second order derivatives in order to get an easy to use test statistic. Small sample performance of the test is studied and compared to other tests by Monte Carlo experiments.     

Shapiro–Francia test compared to other normality test using expected p-value

The Shapiro–Francia (SF) normality test is an important test in statistical modelling. However, little has been done by researchers to compare the performance of this test to other normality tests. This paper therefore measures the performance of the SF and other normality tests by studying the distribution of their p-values. For the purpose of this study, we selected eight well-known normality tests to compare with the SF test: (i) Kolmogorov–Smirnov (KS), (ii) Anderson–Darling (AD), (iii) Cramer von Mises (CM), (iv) Lilliefors (LF), (v) Shapiro–Wilk (SW), (vi) Pearson chi-square (PC), (vii) Jarque– Bera (JB) and (viii) D'Agostino (DA). The distribution of p-values of these normality tests were obtained by generating data from normal distribution and well-known symmetric non-normal distribution at various sample sizes (small, medium and large). Our simulation results showed that the SF normality test was the best test statistic in detecting deviation from normality among the nine tests considered at all sample sizes.     

Normality Test Based on a Truncated Mean Characterization

This article generalizes a characterization based on a truncated mean to include higher truncated moments, and introduces a new normality goodness-of-fit test based on the truncated mean. The test is a weighted integral of the squared distance between the empirical truncated mean and its expectation. A closed form for the test statistic is derived. Assuming known parameters, the mean and the variance of the test are derived under the normality assumption. Moreover, a limiting distribution for the proposed test as well as an approximation are obtained. Also, based on Monte Carlo simulations, the power of the test is evaluated against stable, symmetric, and skewed classes of distributions. The test proves compatibility with prominent tests and shows higher power for a wide range of alternatives.     

Normalized Johnson's transformation one-sample trimmed t for non-normality

The present study suggests the use of the normalized Johnson transformation trimmed t statistic in the one-sample case when the assumption of normality is violated. The performance of the proposed method was evaluated by Monte Carlo simulation, and was compared with the conventional Student t statistic, the trimmed t statistic and the normalized Johnson's transformation untrimmed t statistic respectively. The simulated results indicate that the proposed method can control type I error very well and that its power is greater than the other competitors for various conditions of non-normality. The method can be easily computer programmed and provides an alternative for the conventional t test.